Abstract Generated abstract
This note studies the boundary behavior of regular solutions to uniformly elliptic second order equations in bounded domains whose boundary points admit inner or outer contact by generalized paraboloids. Using barrier constructions and assuming the maximum principle, it establishes lower estimates near boundary minimum points along inward rays, with different rates according to the convergence or divergence of an integral involving the defining function of the paraboloid. Under additional bounds on the lower order coefficients and right hand side, and a tangential first order boundary approximation condition, the paper also proves Lipschitz type control of solutions near arbitrary boundary points. A final result gives an exponentially improved boundary estimate in a model paraboloid when the associated integral diverges.
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UDC 517.955.81
MATHEMATICS
B. N. KHIMCHENKO
ON THE BEHAVIOR OF SOLUTIONS OF ELLIPTIC EQUATIONS NEAR THE BOUNDARY OF A DOMAIN OF TYPE \(A^{(1)}\)
(Presented by Academician A. N. Tikhonov on 8 I 1970)
Let, in a closed bounded domain \((T+\partial T)\) of \(n\)-dimensional Euclidean space \(R^n\), a function \(u(M)\) be defined which is a regular solution of the equation
\[ \sum_{i,k=1}^{n} a_{ik}(M)\frac{\partial^2 u}{\partial x_i \partial x_k} + \sum_{i=1}^{n} b_i(M)\frac{\partial u}{\partial x_i} + c(M)u = f(M). \]
Here the functions \(a_{ik}\) are bounded in \(T\), and
\[ \sum_{i,k=1}^{n} a_{ik}\lambda_i\lambda_k \ge \alpha \sum_{i=1}^{n}\lambda_i^2, \qquad \alpha>0. \]
Let further \(l\) be a ray issuing from the considered point \(M_0\) of the boundary \(\partial T\), such that \(\cos(l,n)>0\), where \(n\) is the inner normal to the surface \(\partial T\) at the point \(M_0\).
Denote by \(\varphi(\rho)\) a function satisfying the conditions:
\[ \varphi(\rho)\in C^{(1)}([0,\rho_0])\cap C^{(\infty)}((0,\rho_0]); \tag{1} \]
\[ \varphi(0)=\varphi'(0)=0; \tag{2} \]
\[ \varphi'(\rho)>0,\qquad \varphi''(\rho)>0; \tag{3} \]
\[ \int_{0}^{\rho_0}\frac{\varphi(t)\,dt}{t^2}<\infty. \tag{4} \]
We shall call a \(\varphi\)-paraboloid (see \((^1)\)) the body \(z_0\ge z\ge \varphi(\rho)\). By \(A_\varphi^*\) and \(A_{\varphi *}\) we denote the classes of domains from \(A^{(1)}\) each boundary point of which can be touched by a \(\varphi\)-paraboloid, respectively from outside and from inside.
The construction of barriers for \(u(M)\), analogous to those considered in \((^2)\), leads to the proof of the following theorems. It is assumed here that the maximum principle is valid.
Theorem 1. If \((T+\partial T)\in A_{\varphi *}^{*}\) and at the point \(M_0\in\partial T\) \(u(M)\) attains its minimum value \(u_0\), then for every ray \(l\) there exists a constant \(c_1>0\) such that, for \(M\in l\) in a neighborhood of \(M_0\),
\[ u(M)-u_0\ge c_1 r_{10}. \]
Here (and below) \(r_{10}\) is the distance from \(M\) to \(M_0\).
Denote by \(\Omega_\varphi^*\) the body \(z\ge -\varphi(\rho)\), \(r\le r_0\) \((r=\sqrt{z^2+\rho^2})\), where \(\varphi(\rho)\) satisfies conditions (1)—(3), but
\[ \int_{0}^{\rho_0}\frac{\varphi(t)\,dt}{t^2}=\infty. \tag{5} \]
Theorem 2. If the function \(u(M)\) is defined in \(\Omega_{\varphi}^{*}\) and at the origin of coordinates \(M_{0}\) attains its minimum value \(u_{0}\), then for each ray \(l\) there exists a constant \(c_{2}>0\) such that, for \(M\in l\) in a neighborhood of \(M_{0}\),
\[ u(M)-u_{0}\geq c_{2}r_{10}\int_{r_{10}}^{r_{0}}\frac{\varphi(t)\,dt}{t^{2}}. \]
Let now \(d(M)\) be the distance from the point \(M\) to the boundary \(\partial T\), and suppose the functions \(b_i(M)\), \(c(M)\), and \(f(M)\) everywhere in \(T\) satisfy the conditions
\[ |b_i(M)|<c_3;\qquad |c(M)|d^{\lambda}(M)<c_3;\qquad |f(M)|d^{\lambda}(M)<c_3 \tag{6} \]
\[ (0<\lambda<1) \]
and, moreover, on the surface \(\partial T\) the function \(u(M)\) satisfies the requirement
\[ \left|u(M)-u_0-\sum_{i=1}^{n-1}a_i x_i\right|\leq c_4\varphi(r_{10}), \tag{7} \]
where \(u_0=u(M_0)\), \(a_i=\partial u(M_0)/\partial x_i\), and for the function \(\varphi(\rho)\) conditions (1)—(4) hold.
Theorem 3. If \(u(M)\) is defined in \((T+\partial T)\in A_{\varphi}^{*}\) and \(M_0\) is an arbitrary fixed point of the boundary \(\partial T\), then for each ray \(l\) there exists \(c_5>0\) such that in a neighborhood of \(M_0\)
\[ |u(M)-u_0|\leq c_5 r_{10}. \]
For harmonic functions, in the case where \((T+\partial T)\in A_{\varphi}^{*}\cap A_{\varphi *}\), the assertion of Theorem 3 follows directly from the results given in \({}^{3}\).
Denote by \(\Omega_{\varphi *}\) the \(\varphi\)-paraboloid for which \(\varphi(\rho)\) satisfies conditions (1)—(3) and (5), and suppose that, for the function \(u(M)\) defined in \(\Omega_{\varphi *}\), on the surface \(z=\varphi(\rho)\) one has
\[ \left|u(M)-u_0-\sum_{i=1}^{n-1}a_i x_i\right| \leq c_6\varphi(r_{10}) \exp\left[-c_7\int_{r_{10}}^{r_0}\frac{\varphi(t)\,dt}{t^2}\right], \]
where \(u_0=u(M_0)\), \(M_0\) is the vertex of the \(\varphi\)-paraboloid, and \(c_6\) and \(c_7\) are fixed positive constants.
Theorem 4. For each ray \(l\) there exist \(c_8\) and \(c_9\) such that, in a neighborhood of \(M_0\),
\[ |u(M)-u_0| \leq c_8 r_{10} \exp\left[-c_9\int_{r_{10}}^{r_0}\frac{\varphi(t)\,dt}{t^2}\right]. \]
Moscow State University
named after M. V. Lomonosov
Received
2 XII 1969
REFERENCES
\({}^{1}\) B. N. Khimchenko, DAN, 192, No. 1 (1970).
\({}^{2}\) B. N. Khimchenko, Differential Equations, 5, No. 10 (1969).
\({}^{3}\) Kjell-Ove Widman, Math. Scand., 21, 17 (1967).